Uraniam U235 has a halflife of approximately 7 Times 108 yea

Uraniam U-235 has a half-life of approximately 7 Times 10^8 years. The decay rate of uranium is proportional to the quantity present. After a nuclear accident, 100 grams of uranium is released into the air. Write down the differential equation describing this system. Find the function describing the amount of uranium remaining after time t, where t is time in years since the nuclear accident. How much uranium is there after two million years? How much is there after two billion years?

Solution

Let the qty of uranium be A(t) gms

a) : decay rate is proprtional to quatity present. where Ao is the quantity present

dA/dt = -k(A) ( -ve sign as the qty decays and \'k\' is a constant)

dA/A = -k*dt

Integrating both sides we get : ln(A) = -kt +c where c is a constant

At t=0; A = Ao ; lnAo = c

Substituting value of c : lnA = -kt +lnAo

-kt = lnA - lnAo

-kt = ln(A/Ao)

b) A = Aoe^(-kt)

In the above equation constant k is the decay constant and k = 0.693/ T_1/2 = 0.693/ 7 x 10^8 =0.099x 10^-8

Ao = 100 gms

A = 100e^(- 0.099x10^-8)

After two million years = 2 x10^6 years

A = 100e^( -0.099x10^-8 x 2x10^6) = 99.8 gms

After two billion years = 2 x 10^9 years

A = 100e^( -0.099x 10^-8 x 2 x10^9)

=82 gms